Weightlessness is a condition in which a body feels no force effects. But that does not mean that there are no forces working. The opposite is the case: A body only then is really weightless when it falls freely under the influence of gravity (a place in space that is completely free of gravitational forces only exists theoretically). Maybe this is difficult to understand, but I try to explain it as simple as possible:
Lets say you are in an elevator standing on a scale. It shows your weight. Now the cable of the elevator tears, and you fall into the void together with the elevator. We will neglect the air friction here. What happens?
Gravity accelerates you and the elevator and the scale with the same rate; in the direction of the ground. But there is no force that acts against the acceleration. Because the scale is falling with you and is accelerated in the same rate, it can't show your weight any more. You are weightless.
That's why not only the astronauts in space in their space ship are weightless, but as well the parachutist in free fall, or you while jumping from the 3-meter board in the pool or from the next best stairs...
Here is a very simple experiment that you can do almost everywhere: Put a heavy book on your hand and jump with it from a chair or other elevated point, and in the moment, when you jump (fall), you won't feel the weight of the book.
Possibilities to create zero gravity on earth
You are always weightless when your body falls freely under the influence of gravity, so it would be enough to just jump up into the air.
However, the duration of the zero gravity condition is very short, and that's why you hardly have a chance to really conciously feel this sensation (but it is possible, e. g. on a trampoline).
But when scientists want to do experiments under zero gravity conditions without flying into space, this time is too short.
One possibility is to build a very high tower, a so-called drop tower, with a tube that is evacuated to minimize air friction and eliminate rest forces, and a device to catch the experiment. This experiment is put into a capsule and dropped in the tower, and at the bottom it is caught by a cushion of styrofoam balls or something similar, that it is not destroyed. The duration of the microgravity is a few seconds, depending on the height of the drop. (If you are interested, there is a formula for calculating the drop height: s = 1/2 g * t² that means the fallen distance corresponds with half the acceleration due to gravity times the duration of the fall in the square)
For some materials science experiments, e. g. special combustion procedures, this time is enough.
But there is one crucial disadvantage: The scientists can't participate and intervene.
For this there is another possibility, the so-called parabolic flights:
If you want to know what that is and how it works, click here.
A further means of parabolic flight is the flight of experiments on sounding rockets. Therefore a rocket is launched to an altitude of between 100 and 800 km, but does not reach orbit and falls back to earth. The experiments are recovered by a parachute, depending on their requirements. Here too the scientists cannot participate, but the microgravity state lasts from 5 to 15 minutes and the experiments can partly be controlled from the ground via telemetry.
Besides, SpaceShip 1, flown by Mike Melvill and Brian Binnie, followed the same trajectory, as will the suborbital spacecrafts intended to take tourists into space in the near future. There are also thoughts about using these flights for science experiments with the possibility for direct interaction by the scientists like on the parabolic flights performed by aircraft.
But why are the astronauts floating weightless in their spaceship ?